Thomas_redesign_ptI
8/10/09
9:20 AM
Page 1
PEARSON’S
Calculus Review, Single Variable
Differentiation
USEFUL DERIVATIVES
DERIVATIVE
The derivative of the function f (x) with respect to the variable
x is the function f ¿ whose value at x is
Limits
LIMIT LAWS
The limit of a rational power of a function is that power of the
limit of the function, provided the latter is a real number.
If L, M, c, and k are real numbers and
lim f (x) = L and
x: c
1.
7.
lim g(x) = M, then
x:c
Sum Rule: lim ( f (x) + g(x)) = L + M
8.
x: c
USEFUL LIMITS
1.
Product Rule: lim ( f (x) – g (x)) = L – M
x: c
2.
FINDING THE TANGENT TO THE CURVE y = f (x) AT
(x0, y0)
Quotient Rule: lim
x:c
Calculate f (x0) and f (x0 + h).
2.
f (x0 + h) - f (x0)
Calculate the slope f ¿(x0) = m = lim
.
h
h :0
3.
If the limit exists, the tangent line is y = y0 + m(x - x0).
x: q
q
E - q(n(neven)
odd) ,
lim
x: ; q
1
= 0
xn
1.
Constant Rule: If f (x) = c (c constant), then f ¿(x) = 0.
2.
Power Rule: If r is a real number,
3.
d
Constant Multiple Rule:
(c – f (x)) = c – f ¿(x)
dx
4.
Sum Rule:
5.
Product Rule:
(provided an, bm Z 0)
4.
lim ( f (x))
x: c
1
+ q (n even)
lim
n = E ; q (n odd)
x: c ; (x - c)
6.
Power Rule: If r and s are integers with no common factor
r, s Z 0, then
r/s
For integers n, m > 0,
5.
lim
x: 0
sin x
sin kx
= k (k constant),
lim x
x = 1, x:
0
7.
r/s
= L
cos x - 1
= 0
lim
x
x: 0
r/s
provided that L is a real number. (If s is even, we assume that
L 7 0.)
d r
x = rx r - 1
dx
d
[ f (x) ; g(x)] = f ¿(x) ; g¿(x)
dx
d
[ f (x)g(x)] = f (x)g¿(x) + f ¿(x)g(x)
dx
g(x) f ¿(x) - f (x)g¿(x)
d f (x)
Quotient Rule:
c
d =
dx g(x)
[g(x)]2
dy du
d
Chain Rule:
[ f (g(x))] = f ¿(g(x)) – g¿(x) =
–
dx
du dx
if y = f (u) and u = g(x)
2.
d
(cos x) = -sin x
dx
4.
d
(cot x) = - csc2 x
dx
5.
d
(sec x) = sec x tan x
dx
6.
d
(csc x) = - csc x cot x
dx
7.
d x
(e ) = e x
dx
8.
d x
(a ) = (ln a)a x
dx
10.
1
d
(loga x) =
dx
x ln a
12.
d
-1
(cos-1 x) =
dx
11 - x 2
14.
-1
d
(cot-1 x) =
dx
1 + x2
1
d
(ln x) = x
dx
11.
d
1
(sin-1 x) =
dx
11 - x 2
13.
1
d
(tan-1 x) =
dx
1 + x2
DIFFERENTIATION RULES
an x n + an - 1x n - 1 + % + a0
an
lim
= – lim x n - m
bm x: ; q
x: ; q bm x m + bm - 1x m - 1 + % + b0
f (x)
L
, M Z 0
=
g(x)
M
d
(tan x) = sec2 x
dx
9.
1.
x:c
3.
3.
lim k = k, (k constant)
x: ; q
For an integer n > 0, lim xn = q ,
lim xn =
The limit of a quotient of two functions is the quotient of their
limits, provided the limit of the denominator is not zero.
6.
lim k = k,
x: c
x: - q
Constant Multiple Rule: lim (k – f (x)) = k – L
The limit of a constant times a function is the constant times the
limit of the function.
5.
f (z) - f (x)
z - x .
z:x
provided the limit exists; equivalently f ¿(x) = lim
x: c
The limit of a product of two function is the product of their
limits.
4.
If P(x) and Q(x) are polynomials and Q(c) Z 0, then the rational
P(x)
P(x)
P(c)
=
function
has lim
.
Q(x)
Q(c)
x: c Q(x)
Difference Rule: lim ( f (x) - g(x)) = L - M
The limit of the difference of two functions is the difference of
their limits.
3.
If P(x) = an x n + an - 1x n - 1 + % + a0 is a polynomial then
lim P(x) = P(c) = ancn + an - 1cn - 1 + % + a0 .
x: c
The limit of the sum of two functions is the sum of their limits.
2.
f (x + h) - f (x)
,
f ¿(x) = lim
h
h:0
d
1.
(sin x) = cos x
dx
f (c) exists
(c lies in the domain of f )
If f ¿(x) > 0 at each x H (a, b), then f is increasing on [a, b].
2.
lim f (x) exists
( f has a limit as x : c)
If f ¿(x) < 0 at each x H (a, b), then f is decreasing on [a, b].
3.
x :c
lim f (x) = f (c)
x :c
9 0 0 0 0
2.
9
1
2.
If f ¿(c) = 0 and f –(c) 7 0, then f has a local minimum at
x = c.
3.
If f ¿(c) = 0 and f –(c) = 0, then the test fails. The function f
may have a local maximum, a local minimum, or neither.
d
-1
(csc - 1 x) =
dx
ƒ x ƒ 1x 2 - 1
If f is continuous on [a, b] then F(x) =
La
dF
d
f (t) dt = f (x),
=
dx
dx La
(2)
19.
20.
d
(coth x) = - csch2 x
dx
21.
d
(sech x) = -sech x tanh x
dx
22.
d
(csch x) = -csch x coth x
dx
INTEGRATION BY PARTS FORMULA
23.
1
d
(sinh-1 x) =
dx
11 + x 2
24.
1
d
(cosh-1 x) =
dx
1x 2 - 1
27.
-1
d
(sech - 1 x) =
dx
x11 - x 2
28.
1
d
(coth - 1 x) =
dx
1 - x2
3.
2.
If f – < 0 on I, the graph of f over I is concave down.
If f –(c) = 0 and the graph of f (x) changes concavity across c then f
has an inflection point at c.
if f ¿ changes from positive to negative at c, then f has a local
maximum at c;
more➤
2
u dv = uv -
du = u + C
L
2.
k du = ku + C (any number k)
L
3.
(du + dv) =
du + dv
L
L
L
4.
L
L
un du =
un+1
+ C
n + 1
(n Z -1)
du
u = ln ƒ u ƒ + C
L
7.
cos u du = sin u + C
L
8.
sec2 u du = tan u + C
L
9.
L
L
17.
cosh u du = sinh u + C
L
18.
u
du
= sin-1 Q a R + C
2
2
1a
u
L
19.
1
u
du
= a tan - 1 Q a R + C
L a2 + u2
20.
du
1
u
= a sec-1 ` a ` + C
L u1u2 - a2
21.
u
du
= sinh-1 Q a R + C (a 7 0)
L 1a2 + u2
22.
u
du
= cosh-1 Q a R + C (u 7 a 7 0)
2
2
1u
a
L
23.
sin u du = -cos u + C
6.
16.
v du
L
1.
10.
780321 608109
La
cot u du = ln ƒ sin u ƒ + C
eu du = eu + C
L
au
15.
audu =
+ C, (a 7 0, a Z 1)
ln
a
L
USEFUL INTEGRATION FORMULAS
INFLECTION POINT
if f ¿ changes from negative to positive at c, then f has a local
minimum at c;
L
14.
f (x) dx = F(b) - F(a).
L
SECOND DERIVATIVE TEST FOR CONCAVITY
If f – > 0 on I, the graph of f over I is concave up.
13.
= - ln ƒ csc u ƒ + C
b
5.
1.
tan u du = - ln ƒ cos u ƒ + C
L
= ln ƒ sec u ƒ + C
If f is continuous at every point of [a, b] and F is any antiderivative of
f on [a, b], then
-1
d
(csch - 1 x) =
dx
ƒ x ƒ 11 + x 2
if f ¿ does not change sign at c (that is, f ¿ is positive on both sides
of c or negative on both sides), then f has no local extremum at
c.
12.
a … x … b.
d
(tanh x) = sech2 x
dx
26.
L
x
(1)
d
(cosh x) = sinh x
dx
1
d
(tanh - 1 x) =
dx
1 - x2
f (t) dt is continuous on
csc u cot u du = -csc u + C
11.
[a, b] and differentiable on (a, b) and
18.
25.
assuming that the limit on the right side exists.
x
d
(sinh x) = cosh x
dx
Suppose that c is a critical point ( f ¿(c) = 0) of a continuous function f
that is differentiable in some open interval containing c, except possibly
at c itself. Moving across c from left to right,
1.
If f ¿(c) = 0 and f –(c) < 0, then f has a local maximum at x = c.
17.
FIRST DERIVATIVE TEST FOR LOCAL EXTREMA
ISBN-13: 978-0-321-60810-9
ISBN-10:
0-321-60810-0
1.
THE FUNDAMENTAL THEOREM OF CALCULUS
Let y = f (x) be twice-differentiable on an interval I.
(the limit equals the function value)
Suppose that f (a) = g(a) = 0, that f and g are differentiable on an
open interval I containing a, and that g¿(x) Z 0 on I if x Z a. Then
f (x)
f ¿(x)
,
= lim
lim
g(x)
x:a
x: a g¿(x)
16.
Suppose that f is continuous on [a, b] and differentiable on (a, b).
1.
Suppose f – is continuous on an open interval that contains x = c.
1
d
(sec-1 x) =
dx
ƒ x ƒ 1x 2 - 1
15.
FIRST DERIVATIVE TEST FOR MONOTONICITY
A function f (x) is continuous at x = c if and only if the following three conditions hold:
L’HÔPITAL’S RULE
Integration
Applications of Derivatives
Continuity
SECOND DERIVATIVE TEST FOR LOCAL EXTREMA
L
sinh u du = cosh u + C
sin2 x dx =
x
sin 2x
+ C
2
4
24.
x
sin 2x
cos2 x dx = +
+ C
2
4
L
25.
sec u du = ln ƒ sec u + tan u ƒ + C
L
csc2 u du = -cot u + C
26.
L
sec u tan u du = sec u + C
L
27.
3
csc u du = - ln ƒ csc u + cot u ƒ + C
1
1
sec3 x dx = sec x tan x + ln ƒ sec x + tan x ƒ + C
2
2
L
Thomas_redesign_ptI
8/10/09
9:20 AM
Page 2
Calculus Review, Single Variable
Volume of Solid of Revolution
b
DISK V =
La
p[ f (x)]2 dx
La
2pxf (x) dx
USEFUL CONVERGENT SEQUENCES
L =
2 [ f ¿(t)]2 + [ g¿(t)]2 dt
Lc
21 + [ g¿(y)]2 dy
1.
2.
where x = f (t), y = g(t)
3.
Numerical Integration
4.
b
f (x) dx L
5.
¢x
(y + 2y1 + 2y2 + % + 2yn - 1 + yn)
2 0
6.
b - a
n and yi = a + i¢x, y0 = a, yn = b
where ¢x =
lim
n: q
gan converges if there is a convergent series gcn with an … cn
for all n 7 N, for some integer N.
(b)
gan diverges if there is a divergent series of nonnegative terms
gdn with an Ú dn for all n 7 N, for some integer N.
ln n
n = 0
lim 1n = 1
n: q
lim x
n
n: q
1.
n: q
2.
x
lim a1 + n b = ex
n: q
xn
n!
3.
= 0
Let sn = a1 + a2 +
an
If lim
= c 7 0, then gan and gbn both converge or both
n : q bn
diverge.
an
If lim
= 0 and gbn converges, then gan converges.
n : q bn
an
If lim
= q and gbn diverges, then gan diverges.
n : q bn
THE RATIO TEST
3.
xn
e x = a , all x
n = 0 n!
4.
q
(-1) x
sin x = a
, all x
n = 0 (2n + 1)!
5.
cos x = a
6.
ln(1 + x) = a
7.
tan - 1 x = a
q
q
Á
La
f (x) dx L
where ¢x =
(a)
¢x
(y + 4y1 + 2y2 + 4y3 + % + 2yn - 2 + 4yn - 1 + yn)
3 0
q
n=0
8.
(b)
b - a
n , n is even and yi = a + i¢x, y0 = a, yn = b
x
, -1 6 x … 1
(-1)n x 2n + 1
,ƒxƒ … 1
2n + 1
m
(Binomial Series) (1 + x)m = 1 + a a bx k, ƒ x ƒ 6 1
k=1 k
n: q
sn exists.
a an converges if nlim
:q
n-1 n
(-1)
n
n=1
m(m - 1)
m
m
,
where a b = m, a b =
1
2
2
q
b
n 2n
(-1) x
, all x
n = 0 (2n)!
q
an Ú 0, lim
+ an.
2n + 1
q
an+1
an = r, then gan converges if r 6 1, diverges if
r 7 1, and the test is inconclusive if r = 1.
SEQUENCE OF PARTIAL SUMS
SIMPSON’S RULE
1
= a (-1)nx n, ƒ x ƒ 6 1
1 + x
n=0
q
= 1 (x 7 0)
lim xn = 0 ( ƒ x ƒ 6 1)
lim
2.
Suppose that an 7 0 and bn 7 0 for all n Ú N (N an integer).
1
n: q
1
= a x n, ƒ x ƒ 6 1
1 - x
n=0
n
LIMIT COMPARISON TEST
n
n
TRAPEZOID RULE
La
(a)
d
b
La
La
LENGTH OF x = g (y)
LENGTH OF PARAMETRIC CURVE
L =
21 + [ f ¿(x)]2 dx
q
1.
Let gan be a series with no negative terms.
0! = 1, 1! = 1, 2! = 1 – 2, 3! = 1 – 2 – 3,
n! = 1 – 2 – 3 – 4 % n
b
L =
COMPARISON TEST
FACTORIAL NOTATION
LENGTH OF y = f (x)
b
SHELL V =
USEFUL SERIES
Infinite Sequences and Series
m(m - 1) % (m - k + 1)
m
a b =
,k Ú 3
k
k!
n=1
q
THE ROOT TEST
sn does not exist.
a an diverges if nlim
:q
an Ú 0, lim 1an = r, then gan converges if r 6 1, diverges if
n
n=1
n: q
GEOMETRIC SERIES
r 7 1, and the test is inconclusive if r = 1.
FOURIER SERIES
ALTERNATING SERIES TEST
1
1
f (x) dx, ak = p
The Fourier Series for f (x) is a0 + a (ak cos kx + bk sin kx) where a0 =
2p L0
k=1
L0
an Ú 0, g(-1)n + 1an converges if an is monotone decreasing and
lim an = 0.
1
bk = p
ABSOLUTE CONVERGENCE TEST
TESTS FOR CONVERGENCE OF INFINITE SERIES
If g ƒ an ƒ converges, then gan converges.
1.
The nth-Term Test: Unless an : 0, the series diverges.
TAYLOR SERIES
2.
Geometric series: gar n converges if ƒ r ƒ 6 1; otherwise it diverges.
Let f be a function with derivatives of all orders throughout some
interval containing a as an interior point. Then the Taylor Series
generated by f at x = a is
3.
p-series: g1n p converges if p 7 1; otherwise it diverges.
2p
q
q
n-1
=
a ar
Polar Coordinates
n=1
LENGTH OF POLAR CURVE
EQUATIONS RELATING POLAR AND
CARTESIAN COORDINATES
x = r cos u
2
2
2
y = r sin u, x + y = r ,
u = tan
-1
b
y
ax b
L =
SLOPE OF THE CURVE r f (u)
La B
r2 + a
b
A =
n: q
an fails to exist or is different from zero.
a an diverges if nlim
:q
n=1
b
1 2
1
r du =
[ f (u)]2du
La 2
La 2
provided dx/du Z 0 at (r, u).
AREA OF A SURFACE OF REVOLUTION OF A POLAR
CURVE (ABOUT x-AXIS)
CONCAVITY OF THE CURVE r f(u)
THE nth-TERM TEST FOR DIVERGENCE
THE INTEGRAL TEST
Let {an} be a sequence of positive terms. Suppose that an = f (n),
where f is a continuous, positive decreasing function for all x Ú N
q
(N a positive integer). Then the series g n = N an and the improper
q
integral 1N f (x) dx both converge or both diverge.
q
p-SERIES
a
k=0
q
d dy
b
a `
du dx (r, u)
d 2y
dx2
=
b
S =
La
2pr sin u r 2 + a
B
dr 2
b du
du
1
a n p converges if p 7 1, diverges if p … 1.
n=1
L0
f (x) sin k x dx, k = 1, 2, 3, Á
f (k)(a)
(x - a)k = f (a) + f ¿(a)(x - a) +
k!
4.
Series with nonnegative terms: Try the Integral Test, Ratio Test, or Root Test. Try comparing to a known series with
the Comparison Test or the Limit Comparison Test.
5.
Series with some negative terms: Does g ƒ an ƒ converge? If yes, so does gan since absolute convergence implies convergence.
6.
Alternating series: gan converges if the series satisfies the conditions of the Alternating Series Test.
If a = 0, we have a Maclaurin Series for f (x).
dx
Z 0 at (r, u).
du
more➤
4
f (x) cos k x dx,
2p
f –(a)
f (n)(a)
(x - a)n + % .
(x - a)2 + % +
2!
n!
f ¿(u) cos u - f (u) sin u
provided
2p
q
dr 2
b du
du
AREA OF REGION BETWEEN ORIGIN
AND POLAR CURVE r = f(u)
f ¿(u) sin u + f (u) cos u
dy
`
=
dx (r, u)
f ¿(u) cos u - f (u) sin u
a
, ƒ r ƒ 6 1, and diverges if ƒ r ƒ Ú 1.
1 - r
5
6